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Finitely generated commutative division semirings. (English) Zbl 1243.16051

Here one-generated commutative division semirings are found. The aim of this note is to find all one-generated (commutative) division semirings. All such semirings turn out to be finite. Keeping this in note some results already available have to be corrected accordingly though the results are fairly basic.
The paper contains the notions of congruence-simple semiring, ideal simple, division semiring, semifield, parasemifield. All semifields and parasemifields are ideal-simple division semirings. Zero multiplication rings of finite odd prime order are both congruence and ideal simple, but not division rings. Every division semiring has at most two ideals. A parasemifield is not one-generated as a semiring. The last six sections contain auxiliary results on commutative semigroups, division semirings – its classifications, a few constructions, division semirings of types II, III, and of type IV, respectively.
Finally the summary reveals together the results discussed. Results on division semirings are such that twelve types of such semirings are commutative. Only three types contain an additively neutral element. Nine types contain a multiplicatively neutral element. Five types contain an additively absorbing element. Seven types of division semirings contain multiplicatively absorbing elements. All division semirings have at most two ideals. Four types are ideal-simple ones among them. Congruence-simple division semirings are of just four types. The last four, respectively, show that (i) finite division semirings are of four types, (ii) ideal-simple commutative semirings are just of five types, (iii) semifields are of seven types, (iv) congruence-simple commutative semirings are just of six types.
Moreover, the following results are noted such as (1) A parasemifield is additively idempotent, provided that it is a finitely generated semiring. (2) A finitely generated division semiring is either almost additively idempotent or it is a finite field or a copy of the semifield for a non-trivial finitely generated commutative group. (3) One-generated division semirings are in particular finite. (4) Every finitely generated congruence-simple commutative semiring is either finite or additively idempotent. (5) A parasemifield is additively idempotent, provided that it is a finitely generated semiring. (6) A finitely generated ideal-simple commutative semiring is either additively idempotent or it is a copy of the semifield \(U(G)\) for an infinite, finitely generated commutative group \(G\). (7) One-generated congruence-simple commutative semirings are just copies of the two element semirings \(Z_1\), \(Z_2\), \(Z_3\), \(Z_4\), finite fields, zero multiplication rings of finite prime order and semifields \(V(G)\), with \(G\) a non trivial finite cyclic group. And all are finite.

MSC:

16Y60 Semirings
12K10 Semifields
16P10 Finite rings and finite-dimensional associative algebras
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